manYPoints – Table of Curves with Many Points
Entry details for q = 31 = 3, g = 4
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Lower bound Nmin = 12

Submitted by Gerrit Oomens
Date 01-01-1900
Reference Jean-Pierre Serre
Quel est le nombre maximum de points rationnels que peut avoir une courbe algébrique de genre g sur un corps fini F_q?
Résumé des Cours de 1983-1984 (=Oeuvres III, No. 132, p. 701-705).
Comments
Tags Explicit curves

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An explicit defining equation     
Isabel Pirsic
06-08-2012 15:57
(y^3-y)*(x^2+1)^2-(x^3-x)

Source:
H. Niederreiter, C. P. Xing, Cyclotomic function fields, Hilbert class fields and global function fields with many rational places, Acta Arithm. 79 (1997), p. 59-76.
Upper bound Nmax = 12

Submitted by Everett Howe
Date 04-14-2010
Reference Jean-Pierre Serre
Rational points on curves over finite fields. With contributions by Everett Howe, Joseph Oesterlé and Christophe Ritzenthaler. Edited by Alp Bassa, Elisa Lorenzo García, Christophe Ritzenthaler and René Schoof
Documents Mathématiques 18, Société Mathématique de France, Paris, 2020
Comments
The Oesterlé bound. Here is a real Weil polynomial that we don't know how to eliminate: x * (x + 2) * (x + 3)^2
Tags Oesterlé bound

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